LINEAR INTEGRO-DIFFERENTIAL EQUATIONS WITH A BOUNDARY · 8. The Self-Adjoint Boundary Conditions....

LINEAR INTEGRO-DIFFERENTIAL EQUATIONS WITH A BOUNDARY

CONDITION*

BY

MINFU TAH HU

CONTENTSPAGE

1. Introduction and Notations. 363

2. The Integro-Differential Equation. 365

3. The Boundary Problem. 368

4. Integro-Linear Independence. 372

5. The Adjoint Integro-Differential Expression. 378

6. A Modified Form for Green's.Theorem. 382

7. The Adjoint System. 3878. The Self-Adjoint Boundary Conditions. 392

9. The Green's Functions. 397

1. Introduction and Notations

It is a well-known fact that linear integral equations of the first and second

kinds may be regarded as the limiting cases, as n becomes infinite, of systems

of n linear algebraic equations in n variables.

The same idea of passing to a limit suggests that one treat the integro-

differential equation

du(x s) re ( s\(A) —~-]-<p(x,s)u(x,s) + J \P^xtju(x,t)dt = \(x,s)

as the limit of a system of n linear differential equations of the first order

of the formf

* Presented to the Society, December 28, 1917. The problem treated in this paper was

first suggested to me by Professor W. A. Hurwitz, to whom, and to Professor M. Bôcher,

I tender my grateful acknowledgment for constant help, suggestions, and criticisms.

t For the system ( o ) when all the equations are homogeneous, a different integro-differential

equation was obtained by Schlesinger (Jahresbericht der Deutschen.Mathe-

matiker-Vereinigung, vol. 24 (1915), p. 84) by means of a process involving

certain changes of the form of the equations (a) before passing to the limit. The equation

thereby obtained differs from ( A ) in that the variable x is complex and all functions involved

are analytic functions in x, that the functions u and X contain another variable r of the same

class as s, and that

tf, ( x, s ) = 0 and X ( x, s ) = <í I ) .\xr J

Trans. Am. Math. Soc. 34 363

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364 minfu tah hu [October

(a)

dill ( X )-"I- 4- hi (x) Ml (X) + •■• +lln(x)un(x) =\i(x),

du ( x}+ lnl(x)uL(x) + ••• +lnn(x)un(x) =\n(x).

dx

We shall have occasion to adjoin to (A) a boundary condition of the type

a(s)u(a,s) + ß(s)u(b,s) + J [A(s, r)u(a, r)

+ B (s, r)u(b,r)]dr = y(s).

This we shall call a two-point boundary condition since it involves the two

values a and b of the variable x. This is obviously the limiting form of the

system of linear boundary conditions usually attached to the finite differ-

ential system (a), as we let the number of equations increase indefinitely.

Throughout this paper, all variables entering will be real. These variables

may be conveniently divided into two classes corresponding respectively to

the first and second arguments of the unknown function u in the equations

(A), (B). The first class of variables is denoted by such letters as x, y, z,

¡z, V, f, and they take on the values in the closed interval

7: a Si x Si b.

We shall speak of this in the future simply as the interval Ix, the subscript x

indicating the variable referred to.

The second class of variables is usually denoted by the letters s ,t,r,a ,t , p,

which take on the values in the interval

J: a Si s Si ß.

In the case of functions of two or more variables, each of which is confined

to one of the intervals 7 and J, we interpret the different variables as rect-

angular coordinates. For instance, the unknown function u(x,s) will be

supposed to be defined in the rectangle

IxJs- a Si x = b, aSisSijS.

In case the variables belong to the same class, we shall have square regions

Ixy or Jal. Likewise, for functions involving more than two variables we have

Then Schlesinger considered also the associated homogeneous equation of the type ( A )

whose solutions are made dependent on the solutions of the former equation. These equations

were also treated in a similar fashion in the notation of general analysis by T. H. Hildebrandt

(these Transactions, vol. 18 (1917), p. 73). [After the manuscript of the present

paper was in the hands of the editors of the Transactions, I was informed by them

that a second paper by Hildebrandt was to appear shortly in the Transactions. See

vol. 19 (1918), p. 97.]


1918] LINEAR INTEGRO-DIFFERENTIAL EQUATIONS 365

such regions as Ixy J., Ixy Jst, etc. All these intervals and regions will be

understood to be closed.

To simplify our work, we shall assume, unless otherwise stated, that all

functions considered are real and continuous (and therefore bounded) in the

respective regions in which they are defined. By a solution of the equations

( A), (B), or any other equation under consideration, we understand, with-

out further specification, a continuous function. A solution of the equa-

tions (A), ( B ), possesses a continuous first derivative with respect to its

first argument. A solution which is identically zero will be termed a trivial

solution.

2. The Integro-Differential Equation

The integro-differential equation

du(x s) Cß Í s\(A) \' +<p(x,s)u(x,s)+J 4<{xt)u(x,t)dt=-k(x,s)

may be reduced, by means of the transformation*

. . -f\(Z,s)dlt . .u(x, s) = e Jv v(x, s),

where y is regarded as a fixed point in Ix, to the equation

dv(x,s) Cß ( s\g-f* Mi, t) - „({, s)]dt(1) àx Ja *\xt)e

Xv(x,t)dt = e^y X (x, s).

This equation is the special case of ( A ) in which the second term of the first

member is lacking. Let us write for convenience

(2)

so that

*(r)*(r)-*(r)-Changing x in (1) into £ and integrating from y to x, we find

v(x,s) =v(y,s) +fR(j.Sy(ï,s)dï

-íT'GíMrMí')'«-«*** This was pointed out to me by Professor Birkhoff.


366 MINFU TAH HU [October

We may now transform back to«(z,s), getting

+jrr[-Ä(r)*(»;)]'(t-,)*«-This is a special case of the equation

(4) «(*,*) =f(x,s) + fp(^)«(^0^.Let us then consider (4).

The function 6 ( f \ ) (called the kernel of the equation) will be supposed

to be continuous in I^Jet- In its appearance, the equation is intermediate

between the Volterra and the Fredholm types; but it behaves like an equa-

tion of the Volterra type because of the variable limit of the first integral.

Since Volterra's method may be applied almost word for word,* we shall

give here only the results.

We are led by the method of successive substitutions to the consideration

of the series

(5)

where

*(ï;)+*(ï;)+*(;:)+-'

<V.)-(Vi)-^;H"IV'(::M'J)«<-

The series (5) converges absolutely and uniformly in Ixi J,t, thus representing

a bounded continuous function, 9 ( ¿ \ ), which shall be called the resolvent

function of the kernel 0 ( | ', ).

The kernel and the resolvent function satisfy the resolvent formulae

<*> •(r.)-(;0+nM;:M«;K» •(;î)-(;i)+jrr'(::)-(!ïK

We now readily establish the

* See Volterra: Leçons sur les équations intégrales, p. 74, where an equation is treated which

is identical with (4), except that ß is replaced by the variable s. The possibility of using the

same method when one of the upper limits is constant was pointed out to me by Professor

W. A. Hurwitz.



Lemma. If d (| î ) is continuous in Ix¡ Jst and f(x, s) is continuous in

Ix J,, then the equation (4) has one and only one solution, namely

(8) u(x,s)=f(x,s)+f£o(X¡.St^f(i¡,t)dtdt.

Returning now to the equation (A'), let

(9)

/(*,*) =R(XyS)u(y,s)+fXR(X¡tay(t,s)d!¡,

where y is regarded as a fixed point in Ix. Thus we obtain by the lemma, for

each assigned function u(y,s), a unique solution of (A') or (-4), which,

if we let

•(;o-r»(;o-(i*)*may be put into the form

u(x,s) = R(AXySJu(y,s)+ jT s[**tJu(y,t)dt

(11)

+fKr)x(*'5)+fKr<)xu'H^Hence, we have

Theorem I. The integro-differential equation (A) possesses one and only

one solution which reduces to the assigned initial function u(y, s) at the fixed

point y in Ix; this solution is given by the formula (11).

Corollary I. If the integro-differential equation (A) is homogeneous, i. e.,

if\(x,s)=0,the solution has the form

(12) u(x,s) = R^XySû(y,s) + £S {^^ u (y, t) dt.

Corollary II. The function

(i3> "(r)-JiKr)x<f'',+fs({Ox(f-H«is a particular solution of the non-homogeneous equation (A), corresponding

to the initial function u(y s) = 0.

Observe that the integrand of the expression w(xy'), when regarded as a

function of x and s, is a solution of the homogeneous equation (A) for each

constant value of £. Thus the particular solution w ( xy * ) of the non-homo-


368 MINFU TAH hu [October

geneous equation (A) is built up from the solutions of the homogeneous

equation by an integration. It is clear that every other solution of the non-

homogeneous equation is obtainable by adding to the particular solution w a

solution of the homogeneous equation.

We shall also have occasion to apply the following:

Corollary III. The function Q ( y \ ), when regarded as a function in x

and s ,is a solution of the homogeneous equation (A), corresponding to the initial

function Q(y't) = - \¡/(y't) at y.

On account of the resolvent formula (6), we have

<"> 8(;î)-(;î)'

and, on account of the first formula (9) and formula (2),

•(;:)--*(.:)■

Consequently, by combining (9), (14), (15),

and because of (10) the equation (6) becomes

which is a solution by Corollary I.

3. The Boundary Problem

Let us now take a linear integral boundary expression of the following type :

U[u] m a(s)u(a, s) + ß(s)u(b, s)

(1) rß+ J [A(s,r)u(a,r) + B (s, r)u(b, r)]dr,

where a(s), ß(s) are continuous functions in J,, A(s,r), B(s,r) are

continuous functions in J,t, and a, b are the end points of the interval Ix.

Let us write from now on

du(x s) Cß f s\(2) L[u]m -^1 + 4>(x,s)u(x,s) + J t{xt)u(x,t)dt.

We shall consider the integro-differential boundary problems



(A) L[u]=\(x,s), (B) U[u] = y(s)and

(Ao) L[u] = 0, (Bo) U[u] = 0.

It has been seen that all solutions of the non-homogeneous and the homo-

geneous equations (A), (A0) are of the forms

(3) u(x,s) = ™(**) + R^XySû(y, s) + £s(yj)«(y ,t)dt,

(4) u(x,s) =R Q5j u(y,s)+ jT S QjJ «(»,*)#,

respectively, where ?/ is a fixed point in the interval Ix at which the initial

function u(y,s) is to be assigned. Both y and the coritinuous function

u(y, s) are arbitrary. But in order to satisfy the boundary condition, it is

clear that the initial function must be suitably chosen.

Substituting in (1) the value of u (x, s) from (3), we find that the boundary

condition ( B ) reduces to

(5) 9(y,s)u(y,s)+ fy(ySt}u(y,t)dt = y(s) ~^ [»(**)]>

g(y,s) =a(s)R(yS} + ß(s)R(bySy

B(,i)-^-><(;')+'<-«j*(")+"['(;i)]-

This is an integral equation for determining the initial function u(y,s).

Likewise, the equation (B0) of the homogeneous system reduces to the

homogeneous integral equation

where

(6)

(7) g(y,s)u(y,s)+ jf G^ySf J u(y, t)dt = 0.

Now we impose the further condition that a(s) and ß(s) be such that

g ( y, * ) do not vanish at any point of J., so that the equations (5) and

(7) may be reduced to integral equations of the second kind. Let us,

then, examine this condition a little further by allowing the point y in the

expression g(y,s) to vary in Iy. Now

g(y,s) = a(s)e Jv +ß(s)e Jy

If g ( y, s ) 4= 0 for a particular value of y, then

a(s) + ß(s)e Ja +0,



i. e.,

(C) a(s) + ß(s)R^baS^ + 0.

Conversely, if (C) is fulfilled, then we shall have g(y, s) + 0 throughout J,

for each value of y in 7„. Thus the condition (C) and the condition g(y, s)

4= 0 are equivalent conditions, but it should be noticed that condition (C)

does not involve y. Hereafter we shall always assume that (C) is fulfilled.

Under the condition ( C ) the equations (5) and (7) become

(5') u(y,s) = F(y,s)+£ K^ySû(y,t)dt,

(7') u(y,s)=f"Kû(y,t)dt,

where

GI a \

K

(8)

( S)=-\ytj g(y,s) '

F(y,s) =g(y,s)

The problem of solving the system (A, B) or (Ao, B0) then reduces to the

determination of the initial function u(y, s) from the equation (5') or (7').

The initial function so determined will give the solution of the system upon

substituting into the equation (3) or (4).

As in the theory of differential equations, the homogeneous system (Ao, B0)

is said to be incompatible if it possesses no non-trivial solution; it is said to

have compatibility of the kth order or index k if there are precisely k linearly

independent solutions.

Suppose ui(x, s), • • •, un(x, s) are linearly dependent solutions of the

homogeneous system (Ao, Bo). Then there exist constants ci, • • •, c„, not

all zero, such that

CiUi(x, s) + ••■ + cnun(x,s) =0

identically in 7S Ja; in particular,

CiUi(y,s) + ■■• + cnun(y,s) =0

for a particular value y in Ix. Conversely, if the initial functions ut(y, s)

are linearly dependent, the solutions of the system formed by means of (4)

will be linearly dependent. Hence

Theorem I. Under the condition (C), a necessary and sufficient condition



that the homogeneous system (Ao, Bo) have index k is that the integral equation

(7') have index* k.

Theorem I'. A necessary and sufficient condition that the system (Ao, B0),

subject to the condition ( C ), be incompatible is that the Fredholm determinant

D (y) of the kernel 7£ ( „Î ) of the equation (T) does not vanish^; if D(y) van-

ishes, the index of the system is finite.

Since the condition ( C ) does not depend on y, the kernel K ( v ", ) of (5')

and (7') and the Fredholm determinant D (y) exist for every y in 7„. Conse-

quently, if D (yo) 4= 0, the homogeneous system (Ao, B0) will be incom-

patible and therefore the equation (7') can have only the trivial solution

u ( y, s ) m 0 for every y. Hence

Theorem II. The Fredholm determinant, D (y), of the equations (5') and

(7') either vanishes everywhere in 7„ or else vanishes nowhere.

Every solution of (Ao, B0) becomes, when x is changed into y, a solution

of (7'). Conversely, however, a solution of (7') will not, in general, when y

is changed to x, become a solution of (Ao, Bo), as is seen by the fact that a

solution of (7') may be multiplied by an arbitrary function of y while a solution

of ( Ao, Bo ) cannot be multiplied by an arbitrary function of x. -Consequently,

(Ao, Bo) is not, in general, equivalent to (7'). It is, however, equivalent in

the special case in which (A0, B0) is incompatible, since then (7') also has

no solution except zero. In this case (A, B) has one and only one solution,

which must, therefore, be the unique solution of (5'). Hence

Theorem III. The systems (A, B), (A0, B0) are respectively equivalent

to the integral equations (5'), (7') whenever the homogeneous system (Ao, Bo)

is incompatible; when (Ao, B0) is compatible, they are equivalent to (5'), (7')

together with the auxiliary equations (3), (4) respectively.

Corollary. When the homogeneous system (Ao, B0) is incompatible, the

non-homogeneous system (A , B) has a unique solution, which is given by

(9) u(x,s) =F(x,s)+f q^xS^F(x,t)dt,

where Q ( x \ ) is the resolvent function of the kernel K(x') of the equations

(5'), (7'). When the homogeneous system is compatible, the non-homogeneous

system (A, B) possesses solutions if and only if

(10) f 4>i(x,s)F(x,s)ds = 0

for all solutions (pi (x, s) of the equation

(11) cp(x,s) =£<p(x,t)K^xtsSj dt.

* Bôcher: An Introduction to the Study of Integral Equations, p. 45.

t Here the quantity y is regarded as a constant, but arbitrary.


372 minfu tah hu [October

4. Integro-Linear Independence

A linear integral expression

U[u; s] = a(s)ui(s) + ß(s)u2(s)

(1)+ j [A(s,r)ui(r)+B(s,r)u2(r)]dr

is said to be integro-linearly self-dependent in the interval Je (or simply self-

dependent) if there exists a continuous function c(s) in J„, not identically

zero, such that

(2) f c(s)Tl[u;s\ds = 0

for every pair of continuous functions ui(s), u2(s); otherwise, it is said

to be self-independent. Two linear integral expressions ¡7i, U2 of the type (1)

are said to be integro-linearly dependent (or simply dependent) if there exist

continuous functions ci(s), c2(s), not both identically zero, such that

(3) j (d(s)Ui[u;s]+ci(s)Ui[u;s])ds = 0

for every pair of continuous functions «i (s) and Ui(s); otherwise they are

said to be independent.

The above definitions of independence and self-independence are obviously

a generalization of the notion of linear independence for a system of algebraic

expressions. We shall derive some necessary and sufficient conditions for

such dependence.

Theorem I. A necessary and sufficient condition that the expression (1) be

self-dependent is that the equations

a(s)c(s) + I c(r)A (r, s)dr = 0,

(4)

ß(s)c(s) + I c(r)B(r,s)dr = 0

have a non-trivial solution c(s) in common. This function c(s) then satis-

fies (2), and conversely every function c(s) which satisfies (2) also satisfies (4).

The theorem is an immediate result of (2) when we observe that Ui(s)

and Ui(s) are arbitrary functions.

Corollary. A sufficient condition that U[u; s] be self-independent is that

either one of the equations (4) possess no non-trivial solution.

Theorem II. When the U of formula (1), § 3, is self-dependent, the homo-

geneous system (A0, B0) of § 3, subject to the condition (C) throughout Js as

considered, has always a non-trivial solution.



By hypothesis there exists a continuous function c(s), not identically

zero in Ja, which satisfies (2) and forms a common solution of the equations

(4). Multiplying (4) by R(y') and R(l') respectively and adding the

results together, we have, by (6), § 3,

(5) ?(y,*)c(*)+jrflc(r)JG(yj)-^[s(^)]Jdr = 0,

and this, by (8), § 3 and by (2) reduces to

g(y,s)c(s) = J c(r)g(y,r)KÍr)dr.

As a solution of (7'), § 3, we have, then,

u(y,s) = g(y,s)c(s).

Consequently the system (Ao, B0) has a non-trivial solution.

Theorem III. 7/ the homogeneous system (A0, B0), subject to the condition

( C ), is compatible and if the expression U is self-independent, not every semi-

homogeneous system

(A,Bo) L[u]=\(x,s), U[u] = 0

possesses a solution.

Suppose the system (A, Bo) does possess a solution for every \(x, s).

Then, by the Corollary to Theorem III, § 3,

f 4>i(y,s)F(y,s)ds = 0J*

for every solution of the equation (11), § 3. This equation reduces to

J. 9(y,s) L \y )\

because of (8), § 3. Expanding U and substituting for w (xy' ) its value from

(13), § 2, we find

f£[R(KaçS)*Ay,s)+£s(açSt)*i(yA)dt]\(ii,s)dsdï

+££[R(\S)*2(y,s)+£s(b^$2(y,t)dt]\(i¡,s)dsd£ = 0,

where

x. r ^ t s<t>i(y,s) Cß(pi(y, r)$i{y> s) = a(s) — j-.-+ I —,-r A(r,s)dr,

" g(y,s) Ja g(y,r)

*, t ^ at N^'(y>g) , rß<t>i(y>r) vt \J*(*.•> = «•> 7(^70+ J. g(y77)-Bir's)dr-



We now find the following equations by assuming first that X ( £, s ) is

zero when !■ = y while it is still arbitrary when £ < y, and secondly that it is

zero when £ Si y and arbitrary when £ > y:

Ä(r)*l(2/'s) +£s{ll)^(y>^dt = ° (a=*=^>

R(bcS^$i(y,s)+fy(bj.tsS)$i(y,t)dt = 0 (i/=g*â&).

Letting £ = a in the first equation and £ = fc in the second, we obtain

#i(y,*)=0, *8(y,*)=0.

Consequently [ c6,- (?/, *) ]/[ £7 (y, s) ] is a solution of (4) which does not van-

ish identically since (pi may be assumed not to be identically zero. There-

fore, by Theorem I, U is self-dependent, which is contrary to hypothesis.

Theorem IV. A necessary condition that two linear expressions U\, Ui of

the type (1) be independent is that each expression be self-independent.

Theorem V. A necessary and sufficient condition that the self-independent

expressions Ui, Ui be dependent on one another is that the equations

ai(s)c2(s) + a2(s)ci(s) + j [c2 (r)Ai(r, s)

+ Ci(r)A2(r,s)]dr = 0,(6)

ßi(s)c2(s) +ß2(s)ci(s) + f [d(r)Bi(r,s)

+ ci(r)B2(r,s)]dr = 0

possess a common non-trivial solution Ci(s), c2(ä). These functions Ci(s),

Ci(s) then satisfy (3); and conversely every pair of functions Ci (s), c2 (s)

which satisfy (3) also satisfy (6).

These theorems follow immediately from the definitions of dependence

and independence so that no proof will be needed.

Let us now consider, more in detail, the case in which, for every value of

* in /,,

ai(s) 0:2(5)

ßi(s) ßi(s)(D) A(s)=-

Equations (6) may now be reduced to

+ 0.

(7)

where

(8)

"»(*)= f [Kn(s,t)ci(t) +Ka(s,t)ci(t)]dt (¿ = 1,2),

Kij(s,t)A(s)

ai(s) ^3_y(í,5)

ßi(s) B^j(t,s)



We shall call the set of- functions

Kn(s,t), Ku(s,t),

(9)Kn(s,t), Kn(s,t)

the kernel-system of (7;.

By the side of (7) we consider the associated non-homogeneous systeni

(10) Ci(s) =fi(s) + J [Kii(s,t)ci(t) + Kn(s,t)a(t)]dt (¿ = 1,2).

We define with Fredholm two new intervals

J?\ aiSisSip\; Jf: «2SisSid2,

such that «i < ßi Si «2 < ßi,

ßi — ai = ßi — a2 = ß — a.

Let J', denote the combined interval of dY1 and J(,2), so that

fF(s)ds= f F(s)ds+ f F(s)ds.Jj> JjiD Jj{2)

We have also four square regions J,\,s>(i,j = \,2)to consider, and J',t

will be used to indicate the totality of all the squares. Then we will map

our functions into the new intervals and regions in such a way that the func-

tions Ka composing the kernel system each occupy one of the four squares.

That is, we define

K(s, t) = Ka(s — ai + a, t — a, + a) for J\{,

(11) f(s) =fi.(s-ai + a) for J\,

c(s) = d(s — ai + a) for J\ (i,j = 1,2).

According to this notation, the equations (11) have the form

(12) c(s) =f(s)+ Í K(s,t)c(t)dt.Jj-

This equation may be treated as an ordinary Fredholm equation by forming

the Fredholm determinant, A, and first minor, A (s, t), of the kernel K(s,t)

in the usual way. We shall call A the determinant of the kernel-system (9).

If A 4= 0, the resolvent function

Q(s, t) = —^—,


376 MINFU TAH hu [October

satisfies the relations

Q(s,t) =K(s,t)+ fQ(s,a)K(<r,t)da,•Jj'

(13)

Q(s,t) =K(s,t)+ f K(s,a)Q((x,t)d(x;•Jj>

and the equation (12) has one and only one solution,

(*) =/(*)+ fQ(s,t)f(t),(14) c(s) =/(*)+ Q(s,t)f(t)dtJj>

Returning to the old coordinates, we can define

Qa (s,t) =Q(s -a + ai, t-a + a,) (for J\{).

This is called the resolvent-system of the kernel-system 7i,y. Equations (13)

become

Qij(s,t) =Kij(s,t)+J [Qn(s,a)Kij(a,t)

nKs + Qi2(s,o-)K2j(a,t)]do-,

Qij(s,t) = Kij(s,t) +J [Kii(s,a)Qij(a,t)

+ Ki2(s, o-)Qij(o-, t)]d(r,and the solution (14) takes the form

r"(16) d(s) =f(s) + [Qn(s,t)fi(t) +Qa(s,t)fi(t)]dt (¿ = l,2).

We now easily infer the truth of the following lemmas:

Lemma I. A necessary and sufficient condition that the system (10) possess a

unique solution is that A 4= 0. If this condition is satisfied, the solution is given

by formula (16); and, in particular, the trivial solution will be the only solution

of the homogeneous system (7).

Lemma II. When A = 0, the system (7) always possesses non-trivial solu-

tions; and a necessary and sufficient condition that the system (10) have solutions

is that the equation

(17) j^(s)f(s)ds= j WAs)fi(s)+ti(s)fi(s)]ds = 0

be satisfied by every solution \p ( s ) of the equation

(18) xft(s) = ft(t)K(t,s)dt.Jj>

By combining Lemma I and Theorem V we have



Theorem VI. If Ui, U2 are self-independent and fulfill the condition (D)

for every value of ß in J8, then a necessary and sufficient condition that they be

independent of each other is that the Fredholm determinant A of the kernel-system

(9) be different from zero.

Theorem VII. If Ui, U2 fulfill the condition (D) ,the equations

(19) Ui = (pi(s), U2 = (pi(s),

when regarded as equations in Ui(s) and «2 (s), possess a unique solution if

and only if Ui, Ui are independent. This solution is integro-lineqr in (pi and c/>2.

For, if we replace Wi and w2 by

Vi(s) = a2(s)ui(s) + ßi(s)u2(s),

v2(s) = ai(s)ui(s) + ßi(s)Ui(s),

and let/i = c62,/2 = (pi, equations (19) become

Vi(s) =/,•(*) 4- f [vi(r)Kii(r,s)+Vi(r)Ku(r,s)]dr (¿ = 1,2),

or simply

*(*) =/(*) + fv(r)K(r,s)dr.

This equation however has precisely the transposed kernel K(r,s), so that

it has a unique solution when and only when A 4= 0. The second part of

the theorem now follows readily.

Corollary I. If (D) is fulfilled, the homogeneous equations Ui = 0,

Ui = 0 possess non-trivial solutions when and only when Ui, Ui are dependent.

Corollary II. If Ui is such that ai(s), ßi(s) do not vanish together* and

if Ui = 0 admits no non-trivial solution, then every self-independent Ui which

fulfills (D) is independent of Ui.

An important application of this corollary is that for a given self-independent

Î7i in which «i (s), ßi(s) do not vanish together, if there can be found a ¿72

such that ( D ) is satisfied and for which Ï72 = 0 admits no solution other than

the trivial one, then Ui, U2 are independent. Unfortunately, I have as yet

been unable to determine whether such a U2 always exists. We shall have

to leave this important general problem without giving a definite answer.

Instead we shall only show the following fact which includes several important

special cases in which we know U2 can be found.

* Obviously ( D ) cannot be fulfilled if ai ( s ), ßi ( s ) do vanish together. On the other

hand, if ai (s), ßi (s) do not vanish together, there always exist functions <*2 (s), #¡ (s)

such that (D) is fulfilled, for instance a¡ = — ßi, ßi — ai.



Theorem VIII. If for a given self-independent expression

Ui[u] = ax(s)ui(s) + ßi(s)u2(s)

+ J [Ai(s,r)ui(r) + BAs,r)u2(r)]dr,

in which ai (s), ßi(s) do not vanish together, there can be found constants ki,

ki such that ki «i (s), kißi(s) do not vanish together and such that

U[[u] = kiai(s)ui(s) + k2 ßi(s)u2(s)

+ J [kiAy(s,r)ui(r) + k2 Bi(s, r)u2(r)]dr = 0

admits no non-trivial solution, then it is possible to find a Ui such that (D) is

fulfilled and that Ui, U2 are independent.

For, suppose ki, k2 are both different from zero, then the theorem is obvious,

because if we group the constants ki, k2 with the unknown functions Ui(s),

Ui(s) respectively, then U\ will have exactly the same form as Ui so that

they both can have no solution. By Corollary II, a Ui exists.

If k2 = 0, then we must have ki 4= 0, «i ( s ) 4= 0 for every value of s in Js,

and, further, on dividing U[ by ki, the equation

Ui = ax(s)ui(s) + I Ai(s, r)ui(r)dr = 0

has no non-trivial solution ui ( s ). Now if we define

Ui = ai(s)u2(s) + I Ai(s, r)u2(r)dr,Ja

then Ui — 0 will have no non-trivial solution n2 ( s ), so that the equations

Ui = 0, Ui = 0 together will admit only the trivial solution Ui ( s ) = 0,

Ui ( s ) = 0. The same argument will enable us to construct a Ui for the

case ki = 0.

5. The Adjoint Integro-Differential Expression

Definition. The integro-differential expressions

du(x s) CB ( s\L[u] = ' dx'-r-<p(x,s)u(x,s) +J $yxt)u(x,t)dt,

M[v] =- -dVÍ^S) + 4>(x,s)v(x,s) + fy(xl)v(x,t)dt



are said to be adjoint to each other; the equations

(Ao) Z[u] = 0,

(2o) M[v]=0

are called adjoint equations.

If we multiply L[u], M[v] respectively by v (x, s) and u (x, s), integrate

with respect to s, and subtract the results, we find

[v(x, s)L[u] — u(x, s)M[v]\ds = r- I u(x, s)v(x, s)ds,

which may be called Lagrange's Identity. Integrating again, with respect

to x, we have the Green's theorem :

(2)

I I [vL[u] — uM[v]]dsdxJxi Ja

I [u(Xi, s)v(Xi, S) — U ( Xi, S ) V ( Xi, s ) ]

These relations hold for any continuous functions u(x, s) and v (x s), pro-

vided they have continuous first derivatives with respect to x.

Let us write for convenience (A0) in the form

— dv(x i) — rß — ( s\(A'o) -M[v]= ^ ' + (¡>(x,s)v(x,s)+y t{xt)v(x,t)dt = 0.

A dash above a function will be used here consistently to indicate the corre-

sponding function of the adjoint equation. The solution of (.do) may then

be written

(3) v(x,s) =R(XySy(y,s)+f*s(XySty(y,t)dt.

There are important symmetrical relations between the functions

To obtain such relations, let us apply Green's theorem to the solutions of

(.do) and (.40). For such functions, u, v, Green's theorem becomes

(4)Xß rß

u(xi, s)v(xi, s)ds = I u(x2, s)v(x2, s)dsJa

for any pair of values xi, x2 in Ix.

Let x3, Xi be respectively the points at which the initial functions of u, v

Trans. Am. Math. Soc. US



are to be assigned. Then, by (12), § 2, the solutions have the forms

u(x,s)=r{ Ju(x3,s)+J SÍ t Ju(x3, t)dt,

v(x,s)=R~( Jv(Xi,s)+J S( f )v(Xi, t)dt.

Substituting into (4) and regrouping the terms, we find

J f(s)u(x3,s)ds = 0,

where

«•>-Kr)s(:*)-*(r)s(r)K'>

Since the initial function u(x3,s) is arbitrary, we conclude that/(s) = 0.

Moreover, the initial function v ( Xi, s ) is also arbitrary, so that, by the lemma

to be proved presently, we obtain the following identities :

r( XlS)s ( XlS\ + ÏÏ( Xlt) S ( Xlt\ + T SÍ XlT) S ( XlT\dr\X3 ) \Xit) V Xi ) \X3s) Ja \X3S ) \Xit)

(5)

=<r)K:0+K:')s(x;:)

These relations hold identically in IXlXiZzXt J, and Ix,XlX,Xl Jst respectively.

In particular, if we let x = Xi = x3, y = x2 = a;4, we have*

(6) <7)-«(l')' -<7<)-<V.)* The first relation (6), and also the first relation (7), may also be inferred from the defini-

tion of R; see (2), § 2.



Letting x = xi, y = x2 = x3 = a;4, we have

A special case of interest is when L [ u ] is anti-self-adjoint, i. e., when

L [u] m — M [u]. In this case, we must have

(8) <t>(x,s)=o, *(,;)--*(.;).

Consequently, we have

(9) *(;')->• »co-'CO-co-We will now prove the lemma which we have referred to, and which will be

useful again later.

Lemma. If h(s) and H ( s, t) are continuous functions such that

(10) h(s)d>(s) + f H(s,t)(j>(t)dt = 0

for every continuous function d> ( s ), then h(s) =0 and 77 ( s, t ) = 0.

It is sufficient to show h(s0) = 0 when a < s0 < ß, because it will then

follow from the continuity of h that h(s) = 0, and therefore H(st) =0.

Let So be any interior point of the interval J,. Let a particular function

(p(s) be defined as follows :

(p(s) =

Then from (10)

0 for | s — s o I > e,

1 for s = So,

continuous, positive and Si 1 for | s — s0 \ = e.

A(«o)4-l H(s0,t)(p(t)dt = 0.J>Q-t

By the first law of the mean,

*(#o) +2dI(so,ti)(p(ti) =0,

where s0 — e < tx < s0 + e. Let | 77 ( s, t ) \ < M, then

|A(»o)| =2ec/>(ii)|77(5o,tiJ| <2eM;

hence h(s0) = 0.



6. A Modified Form for Green's Theorem*

Let Ui [ u; s], Ui [ u; s] be the two integro-linear forms

Ui[u;s] = ai(s)u(a,s) + ßi(s)u(b, s)

(1) rß+ J [Ai(s,r)u(a,r) + Bi(s,r)u(b,r)]dr (4-1,2)-.

Regarding (1) as equations in u(a, s) and u(b, s), and Ui[u; s], Ui[u; s]

as known functions of s, it is seen (Theorem VII, § 4) that if the condition

(D) is fulfilled, it is possible to solve îor u (a, s), u(b, s) uniquely in terms

of Ui and Ui, provided the forms are independent; and furthermore, that

the unique solution will consist of two integro-linear forms in Ui and U2 of

the same form as (1). In this case, the second member of Green's Theorem

((2), § 5), in which we put X\ = a and x2 = b, thus becomes

I [u(b, s)v(b,s) — u(a, s)v(a, s)]ds

(2) Ja rs

= J (Ui[u;s]V2[v;s] + U2[u;s]Vi[v,s])ds,

where Vi[v; s], V2[v; s] are integro-linear forms inv(a, s) and v(b, s) of

the form

Vi[v; s] = yi(s)v(a, s) + Si(s)v(b, s)

(3) rß+ J [Ci(s,r)v(a,r)+Di(s,r)v(b,r)]dr (<-l,2).

Thus we see that Green's theorem may always be written in the form

I J (v(x, s)L[u] — u(x, s)M[v]dsdx)tA\ J* J*

= J {Ui[u;s]Vi[v;s] + Ui[u;s]Vi[v;s])ds

if Ui and Ui are independent and satisfy condition ( D ).

Now, suppose ( D ) is not satisfied, or that ( D ) is satisfied but Ui and U2

are dependent. Will it be still possible to determine Vi and V2 so that the

identity (2) will hold? Let us find the conditions under which V\, Vi can

be determined so as to satisfy (2).

Assuming that U\, U2 have the form (1) and Vi, V2 the form (3), let us

then determine the continuous functions 7¿, 5¿, d, Di ( i = 1, 2 ) so that

* In connection with §§ 6, 7 see the corresponding developments for differential equa-

tions given in these Transactions by Birkhoff, vol. 9 (1908), p. 373, and Bôcher,

vol. 14 (1913), p. 415.



(2) holds for every set of continuous functions u(a, s), u(b, s), v(a, s),

v(b,s). It may be remarked here that the notations Ui[u;s], Ui[u],

Ui(s), Ui will be used indiscriminately for convenience, the same being

true for F¿.

On substituting (1) in (2) and equating the coefficients of the arbitrary

functions u (a, s) and u (b, s), we obtain

ai(s)V1[v; s] + ai(s)Vi[v; s] + J {Vi[v; r]A2(r, s)

+ Vi[v; r]Ai(r,s))dr = - v(a,s),

(5) ßßi(s)V1[v;s] + ßi(s)Vi[v;s]+ {Vi[v; r]B2(r, s)

Ja

+ V2[v;r]Bi(r,s))dr = v(b,s)

as a necessary and sufficient condition that Ui, U2 defined by (1) should

satisfy (2). Substituting in these equations the expressions for Fi and F2

from (3) and collecting the coefficients of the arbitrary functions v ( a, s )

and v ( b, s ), we find, by the lemma proved at the end of § 5, that the following

identities give a necessary and sufficient condition for Fi, F2 as defined by (3)

to satisfy (5) :

a2(s)yi(s) + ai(s)y2(s) 4-1=0,

ßi(s)yi(s) + ßi(s)y2(s) =0,

a2(s)di(s) + ai(s)ô2(s) =0,

(66)A(*)*i(»)4-l8i(«)í«(*) -1=0,

a2(*)Ci(*,r) + ai(s)d(s,r) + yi(r)A2(r, s)

+ y2(r)Ai(r,s)

+ j {A2(t,s)Ci(t,r)+Ai(t,s)C2(t,r))dt = 0,

(70) ß2(s)Ci(s,r) + ßi(s)C2(s,r) +yi(r)B2(r,s)

+ y2(r)Bi(r,s)

+ j {B2(t,s)Ci(t,r)+Bi(t,s)C2(t,r))dt = 0,



a2(s)Di(s, r) + ai(s)D2(s,r) + ôi(r)A2(r,s)

+ Si(r)Ai(r,s)

(76)

+ j (A2(t,s)Di(t,r) +Ai(t,s)D2(t,r))dt = 0,

ßi(s)Di(s,r) +ßi(s)D2(s,r) + 6i(r)B2(r,s)

+ h(r)Bi(r,s)

+ f (Bi(t,s)DAt,r)+Bi(t,s)D2(t,r))dt = 0.

Thus these eight equations form a necessary and sufficient condition that Z7i,

Ui, Vi, Vi as defined by (1) and (3) satisfy (2). We will now inquire under

what conditions the continuous functions y i, 5,-, C¿, 7)¿ ( i = 1, 2 ) can be

determined to satisfy equations (6a), (66), (7a), (7b).

If A ( s ) 4= 0 for every value of s in J,, there will be a unique solution of

equations (6a) and (66), namely

(8) ô-c-n*-1^. «^-(-d*-1^ «-i'a>-

On the other hand, if for a particular value, s0, we have A(s0) = 0, then,

in order that the matrix and the augmented matrix of the system (6a) have

the same rank, we must have ßi(so) = ßi(so) = 0. But this cannot be

the case, as we see from the second equation (66). Consequently, 7,-, ¿¡,-

cannot be determined when the condition (D) is not fulfilled. The condition

(D) is then a first necessary condition that we have to impose on Ui, U2

in order that the problem in question be possible.

Assuming then that (D) is satisfied by Ui and Ui, let us now consider

the system (7a).

Using the notation (8), § 4 and letting

fi(s,T)=

(9)

7i (r) Ka(s,r)

-72(r) Kn(s,r)

oi(r) Ka(s,r)

-h(r) Kn(8,r)9i(s,r) =

the equations (7a) may readily be reduced to the form

(10) Ci(*'r) =f*<-*> r)+X [^(«'OCiíí.r)

+ Ki2(s,t)Ci(t,r))dt (¿ = l,2).

In this form we have precisely a system of equations of the type (10), § 4.



Now if the Fredholm determinant A of the kernel system Kij (s, t), (i, j = 1,

2 ), is different from zero, we have by Lemma I, § 4, a unique solution of the

equations, which is given by

Ci(s,r) =fi(s,r)+ j [Qii(s,t)fi(t,r) + Qii(s,t)fi(t,r)]dt

(¿ = l,2).

Because of (9) and the resolvent relations (15), § 4, this solution simplifies into

d(s,r) =yi(r) Qa(s,r)

- 7i(r) Qn(s,r)

and because of (8) it further reduces to

1(11) d(s,r) =

A(r)

ßi(r) Q*(s,r)

ßi(r) Q,i(s,r)

Similarly, for the system (7b) we have the unique solution

1(12) Di(s,r) =

A(r)

ai(r) Qa(s,r)

ai(r) Qn(s,r)

(¿ = 1,2),

(¿ = 1,2).

(¿ = 1,2).

On the other hand, if A = 0, solutions of (7a), (7b) both exist by Lemma II,

§ 4, if and only if

£ [M')M',r)+M')M',r)]di = 0,

J [î(s)gi(s,r) +ypi(s)g2(s,r)]ds = 0

for every non-trivial solution, xf'i(s), \f/2 (s), of the equations

(13) ti(s) = j [ti(t)Kii(t,s) +rpi(t)Ka(t,s)]dt (¿ = 1,2).

Suppose both of these conditions are satisfied. Substituting the values of

fi,fi,Çi, Ci from (9) and 71, 72, ii, ^2 from (8), we have

ft(r)A(r)

j [ti(s)Kii(s,r)+ti(s)Kn(s,r)] ds

-ßj^f\ti(s)Kii(s,r)+4,2(s)Kii(s,r)]ds = Q,

?^f\ti(s)Kii(s,r)+ti(s)Kii(s,r)]ds

ai(r)

A(r)f [ti(s)Ku(s,r) +h(s)Kn(s,r)]ds - 0.



These equations may now be regarded as a system of linear algebraic equations

whose determinant, A ( r ), does not vanish for any value of r in JT. Whence

-ß

[M»)Ku(s,r) +*i(s)K2i(s,r)]ds = 0 «-1,2),Íi. e., ipi = 0, fa = 0 because of (13). But this is contrary to the fact that

fa(s), fa(s) are a non-trivial solution of (13). Hence C¿, 7),- cannot be

determined when A = 0. Thus we have A 4 0 as a second necessary con-

dition to be imposed on Ui, U2; that is (§ 4, Theorem VI), Ui, Ui must be

independent in addition to fulfilling the condition ( D ). Hence

Theorem I. A necessary and sufficient condition that the expressions Vi, V2

of the type (3) be determinable so that the identity (2) holds for every set of con-

tinuous functions u(a, s), u(b, s), v(a, s), v(b, s), is that Z7i, U2 fulfill

condition (D) and that they be independent. The determination is unique and

given by formulas (3), (8), (11), (12).

Now let us suppose that we start from the expressions Vi, V2 just deter-

mined and that we try to determine Ui, U2 so as to satisfy (3). We form the

determinants A ( s ), A for the expressions V\, V2 corresponding to the deter-

minants A (s), A for Ui, Ui, and denote by (D) the condition that, for

every value of s in J,, A ( s ) 4 0. Then, by the theorem just stated, since

Ui, Ui do exist, we have the

Corollary I. If Ui, U2 are independent and fulfill the condition ( D ),

then the expressions V\, Vi are also independent and fulfill the condition (D).

Thus in this case the two sets of expressions are uniquely determinable from

each other.

We see that the necessary and sufficient condition of Theorem I is precisely

a necessary and sufficient condition that the system Ui = 0, Ui = 0, admit

no non-trivial solution (Theorem VII, Corollary I, § 4). Hence

Corollary II. If U\, Ui are such that the system Ui = 0, Ui = 0,

admits no non-trivial solution, then V\, Vi can be determined and they are such

that the system V\ = 0, Vi = 0, admits no non-trivial solution.

The following fact will be useful later.

Corollary III. If Vi, Vi exist, then u(a, s), u(b, s) can be uniquely

expressed in terms of Ui and U2 in the form

u(a,s) = - ji(s)Ui(s) - yi(s)Ui(s)

-£[Ui(r)Ci(r,s) + Ui(r)CAr,s)]dr,

(14)i*(6,*) = i,(*)l7i(*) + *i(*)£M*)

»/3

+ J [Ui(r)Di(r,s) + U2(r)Di(r, s)]dr.



The existence of a unique solution follows from Theorem VII, § 4, and

there it is also shown that the solution is integro-linear in Î7i and U2. Thus

we need now only to verify the formulae (14). For this purpose we assume

u(a,s) = a[(s) Ui(s) + a2(s) U2(s)

+ j* [A[ (s., r) Ux(r) + A'2(s, r) U2(r)]dr,

u(b,s) =ß[(s)Ui(s) + ß'2(s)Ui(s)

+ fß [Bi,(s,r)Ui(r) + B'2(s,r)Ui(r)]dr.Ja

Substituting in the expression

i'ßI [u(b,s)v(b,s) — u(a,s)v(a, s)]ds

and collecting the coefficients of Ui and U2, we find

rß| [u(b,s)v(b,s) — u(a,s)v(a,s)]ds

(15)= fß[Ui (s)V2 (s) + U2 (s)V[(s) )ds,

where

V'i(s) = - a'i(s)v(a,s) + ß'i(s)v(b,s)

(16) Cß- J [v(a,r)Ai(r,s) - v(b, r)B{(r, s)]dr.

But (15) is exactly the identity (2). Since we have seen that for each given

set of Ui and U2, the expressions Vi, V2 are uniquely determined and are

given by (3), the expressions (16) and (13) are identical. Hence we have

the formulas (14).

7. The Adjoint System

It has been seen that the expressions Fi, F2 are uniquely determined for

each U2 integro-linearly independent of Ui and fulfilling ( D ). Now let U2

be another expression independent of Ui and let V\, V2 be the corresponding

expressions thereby determined. We are to see how the two sets, of F,- are

related to one another.

The two sets of expressions, Ui, U2, Vi, V2) and Î7i, U'2, V[, V'2, satisfy

the identity (2) of § 6. Consequently,

J {(Ui[u;s]V2[v;s] + U2 [ u; s] Vi [v; s] ) ds

{Ui[u;s]V2[v;s] + U'1[u;s]V'i[v;s])ds.-£


388 MINFU TÄH HU [October

Let UijUi, U'2 be written in their full form and the coefficients of the arbitrary

functions u(a, s), u(b, s) be equated to zero; we obtain

oti(s)Vi[v;s] + oti(s)Vi [v;s]

+ j (Ai(r,s)Vi[v;r] + Ai(r,s)Vi[v;r])dr

= ai(s)V'2[v;s] + a2 (s) V[ [v; s]

+ £ {Ai(r,s)V'2[v;r]+A'2(r,s)V[[v;r])dr,

(1)ßAs)V2[v;s] + ß2(s)V1[v;s]

+ f {Bi(r,s)Vi[v;r] + B2(r,s)Vi[v;r])dr

= ßi(s)V'2[v;s] + ß'2(s)V'l[v;s]

■>ß

+ j {Bi(r,s)V'2[v;r]]+B'2(r,s)V'1[v;r})dr.

Let us denote by 3> [ V; s ], ty [ V; s] respectively the expressions on the right

of these equations. Since A ( s ) 4 0, we find

Vdv.a] =Fi[V';s]+ j {Kn(s,r)Vi[v;r] + Kn(s, r)V2[v; r])

(¿ = 1,2),where

[«,(*) $[V';s]

dr

(- 1Y~1Fi[V';s] = K

ßi(s) *[V';s]A(s)

The expressions Fi, F2 are integro-linear and homogeneous in V[ and V'2.

As A 4 0, these equations may be solved for Vi and V2, and the solution

is unique, having the form

Vi[v; *] = Fi[V; s] + £ (Qn(s,r)Fl[V; r] + Qi2(s, r)F2[V; r))dr

(¿ = 1,2).

The expressions in the second member are obviously integro-linear and homo-

geneous in V'i and V'2, so that these equations may be regarded as an integro-

linear transformation between the expressions Vi, Vi and V[, V2. Upon

simplifications due to the resolvent relations (15) of § 4, these equations take

the final form

Vi[v;s] = MAs)V[[v;s]+£ Ni(s,r)V'i[v;r]dr,

(2) r>V2[v;s] = V'2[v;s] + Mi(s)V'1{v;s]+j N2(s, r)V¡[v; r]dr.



Similar equations may be obtained for expressing Fl, F2 in terms of V\, V2

by solving (1) for V\ ,V2. It is important to notice that both of these integro-

linear transformations are unique, since all the coefficients depend only on

the coefficients of Ui, U2, and U2.

The importance of the equations (2) lies in the fact that Vi is integro-

linear and homogeneous in V\, so that whenever the boundary condition

Fl = 0 is satisfied, the condition Fi = 0 is also satisfied, and vice versa.

For this reason, we may state :

Theorem I. The condition Vi = 0 is essentially determined by the condi-

tion Ui = 0, and conversely.

Definition. A pair of boundary conditions Ui = 0, V\ = 0 are said to

be adjoint to each other if Ui, Vi satisfy a relation of the form (2), § 6, where

<72 is independent of Ui and the condition A (s) 4= 0 is fulfilled. The systems

'Ao) L[u] = 0, (fio) Ui[u] = 0,

(Jo) M[v]=0, (Bo) Fi[«] = 0

are called adjoint systems.

It follows from Theorem VIII, § 4, that an adjoint boundary condition

always exists if the function ai(s) does not vanish in J„ and the Fredholm

determinant of [ — Ax(s, r)]/[ai(s)] is not zero; and also under certain

more general conditions there specified.

As we have done in § 3, we will restrict ourselves to the case in which the

system (A0, B0) is subject to the condition

(C) ai(s)+R^baS^ßi(s) + 0.

If we consider the adjoint system (A0, B0), we find that a similar condition

(C) yi(s)+R^jy^s) + 0

is fulfilled. For, from the formulas (8), § 6 and (6), (7), § 5, we have

ßi(s)+R(^Sâi(s)

7Us)+R[a )oi(s)-A(#)

«i(*)+Ä(a')&(*)

r,.,»(i*)Hence

Theorem II. If the system (A0, B0) fulfills the condition (C), then the

adjoint system (Ao, Bo) fulfills a similar condition (C).



We shall next prove

Theorem III. The adjoint systems (A0,Bo), (Ao,B0), subject to the

condition (C), have the same index.

Let n be the index of the system (¿to, B0) and m that of the adjoint system

(Ao, Bo). Let «i, • • • , m„ and Vi, • • •, vm be respectively complete sets of

linearly independent solutions of the systems. Let u be any solution of the

equation (^4o), and v any solution of (A0).

Applying Green's theorem to u and Vi, we have

(3) f Ui[u;s]V2[Vi-,s]ds = 0 (¿ = 1,2, •••, m)

for all solutions u of (Ao). As before, let y be any fixed value of x at which

the initial function u(y, s) is assigned. Then, by formula (7), § 3,

Ui[u;s] = g(y,s)u(y,s)+J g( \u(y,r)dr,

where g(y,s) and G ( s'r) are given by (6), § 3. Because (3) has to hold

for all continuous functions u(y,s),v/e have

g(y,s)V2[Vi;s]+£v2[vi;r]G^yrsSjdr = 0,

or

g(y,s)V2[vi;s] = J g (y, r)V2[vi; r]Ky \ dr.

That is to say, (pi(y, s) = g(y, ») F2 [ »,•; s], (i = 1,2, • • •, m), are solu-

tions of the equation

(4) d>(y,s) = £ (p(y,r)K^yrs^dr.

On the other hand, the initial functions of Mi, • • •, m„ form a complete set of

linearly independent solutions of the equation

(5) m(y, s) = J k( S\u(y,r)dr

adjoint to (4). Hence if it can be shown (as we will now do) that the func-

tions V2[Vi;s] (i = 1,2, • • • ,m) are linearly independent, then (pi(y, s)

will constitute m linearly independent solutions of (4), and therefore m Si n.

For, suppose V2[ví; s] (¿ = 1,2, • • ■, m) were linearly dependent. Then

there would exist constants Ci, • • •, cm, not all zero, such that

ci Pî[*i; s] + ■■■ + cmV2[vm;s] = 0.Let us define

Vo = Ci Vi + •■• + cmvm.



Then v0 is also a non-trivial solution of the system (A0, B0), and satisfies

V2[v0;s] = ci V2[vi; s] + • • • + cm V2[vm; s] = 0.

But this is contradictory, because V\ = 0, V2 = 0 admit no non-trivial

solution. This completes the proof that m =i n.

In the same manner, the functions fa(y, s) = g(y, s)U2[uí; s] (i = 1,2,

• • • , n ) form n linearly independent solutions of the equation

(6) *(y,s)= £t(y,r)K(KyT^)dr,

where K has the same meaning in the adjoint system (A0, B0) as K has in

the original system (Ao, Ba) - On the other hand, the initial functions of

Vi, • ■ • ,vm form a complete set of linearly independent solutions of the equation

(7) v(y,s)=£ K(xy)v(y,r)dr

adjoint to (6). Hence ntim. When combined with the previous result,

we have m = n. Thus we have established Theorem III and also

Theorem IV. Let ui, • • ■ , un be a complete set of linearly independent

solutions of the system (Ao, B0), and Vi, • • • , vn a complete set of linearly inde-

pendent solutions of the adjoint system (Ao, Bo) - Then the functions

(8) 4>i(y,s) = g(y,s)V2[vi-,s] (i = l, 2, ••-, n)

form a complete set of linearly independent solutions of the equation (4), and the

functions

(9) ti(y,s) =g(y,s)U2[ui;s] (i = 1,2, -••, n)

form a complete set of linearly independent solutions of the equation (6).

If we replace <j>i(y, s) in the equation (10) of § 3 by the values (8), we

obtain from the second part of the Corollary of Theorem III, § 3, the

Theorem V. A necessary and sufficient condition that a non-homogeneous

system ( A , B), subject to the condition ( C), possess a solution when the reduced

system (Ao, Bo) is compatible and when the adjoint system (Ao, Bo) exists,

is that

(10) j F(y,s)g(y,s)V2[Vi;s}ds = 0

for every Vi which satisfies the adjoint system (A0, B0).

By means of (8), § 3, this condition may be given the form

(H) XXT(5) -Ul[W(Xy)j)V^V^^dSS°-



8. The Self-Adjoint Boundary Conditions

We have shown that two different choices of the auxiliary boundary ex-

pression U2 independent of the given Ui and fulfilling the condition (D)

lead to two expressions V\ which are connected by an integro-linear trans-

formation. Furthermore, this transformation is unique in both ways. This

fact is important for us here, because in seeking the conditions that a given

expression Z7i be self-adjoint, it is sufficient to seek the conditions that a par-

ticular Fi thereby determined be connected with U\ by an integro-linear

transformation.* It is clear that if one particular Fi is integro-linearly con-

nected with Ui, then every V\ will be so connected.

Suppose the condition Ui = 0 is self-adjoint, and that, for a particular

choice of U2, we have

(1) Vi[u;s] = M(s)Ui[u;s\ + j N(s, t)Ui[u; t]dt,

where

Ui[u;s] = ai(s)u(a,s) + ßi(s)u(b, s)

+ I [Ai(s, r)u(a,r) + Bx(s, r)u(b, r)]dr,

Vi[u;s] = yi(s)u(a,s) + di(s)u(b, s)

+ j [Ci(s, r)u(a,r) + Di(s, r)u(b, r)]dr,

the functions 71, ôi, Ci, Di having the values given by the equations (8),

(11), and (12) of § 6. The equation (1) may be thrown into the following

form

Vi[u; s] = M (s)ai(s)u(a, s) + M (s) ßi(s)u(b, s)

+ j [M(s)Ai(s,r) + N(s,r)ai(r)

+ 1 N(s,t)Ai(t,r)dt]u(a,r)dr

+ j [M(s)Bi(s,r)+N(s,r)ßi(r)

+ j N(s,t)Bi(t,r)dt]u(b,r)dr,

whereby we obtain

(2) yi(s) =M(s)ai(s),

* Professor D. Jackson takes the same point of view in his article, in these Transac-

tions, vol. 17 (1916), pp. 418-424.



(2') íi(») = Jf(*)i8i(*),

(3) Ci(s«, r) = M(s-)Ai(s,r) + N(s,r)ai(r) +J N (s ,t)Ai(t, r)dt,

(3') Di(s,r) = M(s)Bi(s,r)+N(s,r)ßi(r) + I N(s, t)Bi(t, r)dt.Ja

Equations (2) and (3) may be regarded as determining the functions M

and N; equations (2') and (3') then constitute the conditions which must be

imposed on Ui in order that it be self-adjoint.

Substituting the values from (8), § 6, for 71 and 5i, the equations (2), (2')

becomeA(s)M(s)ai(s) - ßi(s) =0.

ai(s) -A(s)M(s)ßi(s) =0.

Now ai (s) and ßi(s) cannot vanish together since we must have A (s) 4 0

throughout J, in order that the adjoint expressions exist. Hence, we must

haveA(s)M(s) 1

1 A(s)M(s)= 0

for every value of s in J. That is,

M(s) = ±K~]-

It follows that

(4) ai(s) = ±ßi(s) 4 0 (throughout Ja).

Conversely, when (4) is satisfied,

M(s) ■-A(*r

Equation (4) is a first necessary condition.

Assuming then that (4) is satisfied, let us proceed to consider the equations

(3), (3'). These equations are Fredholm equations with the kernels

— [Ai(t, r)]/[ai(r)] and — [Bi(t, r)]/[ßi(r)] respectively. It is con-

ceivable that the Fredholm determinant of either one of these kernels might

be zero. We shall now show that in such case, no self-adjoint system is

possible.

Let us suppose the Fredholm determinant of the kernel — [Ai(t,r)]/

[ ai (r) ] to be zero, and a self-adjoint expression to exist so that the equation

(3) has a solution. For this it is necessary that

(5) Ja L oti(r) ±A(s) ai(r) J



for every non-trivial solution d>(r) of the transposed equation

The ± signs correspond to those of (4). Because of (6), the condition (5)

has the form

(p(s) , CßCi(s,r)+ Ja -aT(rT<t>ir)dr = 0'

or, since

±A(s)

ßi(s) ±ai(s)7i(«) = A(s)~ A(s) '

■fl

(?) ^(s)aiTs)+l *<'''>|£j* = 0.Let us put

i \ *(*) tu \ r>m ( a, s ) = —,—T , m ( o, s ) = 0.ai(s)

Then equations (6) and (7) become

Ui[u;s] = 0, Vi[u;s] = 0.

The identity (2), § 6 now becomes, if we let u (x, s) = v (x, s),

-ami*--whence <p(s) — 0. But this is contrary to the fact that 4>(s) is a non-

trivial solution of (6). Thus we have derived a second necessary condition

for the existence of self-adjoint expressions, namely, that both* — [Ai(t, r)]/

[ «i ( r ) ] and — [Bi(t, r)]/[ßi(r)] have non-vanishing Fredholm deter-

minants. This condition is sufficient to insure the existence of a unique

solution for each of the equations (3), (3'), and we shall have a third necessary

condition upon equating these solutions to each other. It is also clear that

these three necessary conditions combined are also sufficient for the existence

of self-adjoint expressions.

To determine the explicit form of the third condition, it is convenient to

choose a particular U2 which will simplify the computation. We shall choose

for instance f/2 such that

a2(s)=-0, &(i) = l, A2(s,r) =B2(s,r) = 0.

This U2 is integro-linearly independent of the given Ui. To prove this we

* The proof for — [Bi (t, r)] l[ßi(r)] proceeds exactly in the same way.



note that A(s)=ai(s)40in./S; and according to the notations of § 4,

Ku(s,r) = Kn(s,r) - 0,

Ai(r,s) (throughout J„ ) •K22{s>r) = -^Ä~sT>

Consequently K ( s, r ) = 0 whenever the second argument r is in the interval

«7(1). Let us write for short

Ai (r,s) Bi(r,s)A(s'r) = --îJsT' ®{s,r) = -^ÄsT'

and denote their resolvent functions, which by hypothesis exist, by A' ( s, r ),

B'(s,r). Then

i = ¿(î.r<!>, /„..... _n=o n! Jj, Jj, \si - - ■ snJ

= X —-j— I • • • I 7^22 ( / " ) dsi • ■ ■ dsn.n=0 U\ Ja Ja \Si ••■ SnJ

This is different from zero, because it is precisely the Fredholm determinant

of the kernel A ( s, r ). This completes the proof that U2 is integro-linearly

independent of Ui by Theorem VI, § 4.

From the resolvent relations (15), § 4, we have also the following further

facts *

Qn(s,r) = Qn(s,r) = 0,

§22(5, r) = 7^22(5, r) + J Qa(s, t)K22(t, r)dt,

Q22(s, r) = 7^22(5, r) + J K22(s, t)Q22(t, r)dt,

Qu(s,r) =Ki2(s,r) + j Ki2(s, t)Q22(t, r)dt,

whence Q22 ( s, r ) = A' ( s, r ). Furthermore,

n 1 \ Qu(s,r)Ci(s,r) = - ^ , Di(s,r)=0.

If we let ai(s) = ± ßi (s) = 1, as we may do without loss of generality,

we have

A(s) = l, A(s,r) = -Ai(r,s), Ü (s, r) = =F Bi (r, s),

Ku(s,r) = ±U(s,r) =F A(*,r),

-Ci(s,r) = Qu(s,r) = =F A' (s, r) ± M (s, r) ± J U(s ,t) A' (Í, r)dt,

Trans. Am. Math. Soc. 90



and the equations (3), (3') have the following form

N(s,r) = Ci(s,r) ±A(r,s) + f &(r,t)N(s,t)dt,

N(s,r) = ±»(r,*) 4- j U(r,t)N(s,t)dt.

\dt

Solving,

N(s,r) =[Ci(s,r)±A(r,s)]+f A'(r, t) [Cx(s, t) ± A(i, s)]<

= ±rA'(#,r) 4-A'(r,5) + f A' (*, 0 A' (r, t)dt\ =F M (s, r)

T f*®(s,a)[A'(o-,r) +A'(r,a)

+ J A'(o-,t)&'(r,t)dt der,

N(s,r) = ±m(r,s)±f W (r, t)U(t, s)dt = ±W(r,s).

Equating and transposing,

[A'(s,r) 4-A'(r,i) 4- jT A'(*, i)A'(r, *)#] = *(*,0 4-*'(r,*)

4- r*(#,<r)rA'((r,r) 4-A'(r,<r) 4- f A' (a, t) A' (r, f)d*l der.

Upon solving and simplifying, we obtain the last condition in the final form

A'(*,r)4-A'(r,*)4- f A'(#, a) A'(r, a)da(8) Ja Cß

= &(s,r)+W(r,s) + V (s, a)V (r, cr)d<r.

Theorem I. Tirera/ self-adjoint integro-linear boundary condition may be

reduced to the form

U[u; s] = u(a, s) ±u(b, s) — I [m (a, r) A(r, s)

±u(b,r)®(r,s)]dr = 0,

in which the Fredholm determinants of A and S are not zero and their resolvent

functions A' and 8' satisfy the relation (8). Conversely, every condition of

this form is self-adjoint, provided, of course, that the condition (C)

i*i(¡') + .

is fulfilled.



Corollary. When the integro-differential expression L [ u ] is anti-self-adjoint

and the boundary condition Ui [ u ] =0 is self-adjoint, the latter must have the

form

U[u] = u(a, s) + u(b, s) — I [u(a, r)A(r, s)

+ u(b, r)U(r, s)] dr = 0.

For, when L[u] is anti-self-adjoint, R(y') = 1 ; so the condition (C) will

not be satisfied when a(s) = — ß(s) = 1.

9. The Green's Functions

In the theory of linear differential equations the conception of the Green's

functions enables us to write down in an explicit form the solution of a semi-

homogeneous boundary problem consisting of a single linear differential equa-

tion of the nth order, or a system of n linear differential equations of the first

order, and of a system of n homogeneous linear boundary equations, whenever

the reduced system is incompatible.*

Following out this analogy, we are led to try to find a solution of a system

(A,Bo) L[u]=\(x,s), U[u]=0

in the form

(1) u(x,s) =jy(yS^\(y,s)dy+fa£G(XSt}\(y,t)dtdy,

where 77 and 6? are independent of X. These two functions we shall call the

system of Green's functions for (A, Bo) - We may arrive at such functions

by imposing certain conditions of discontinuity suggested by the discon-

tinuities of Green's functions for differential equations.

Let G ( J ', ) be continuous, together with its first partial derivative with

respect to x, throughout the region IxyJst; let 77 (**) be continuous, to-

gether with its first partial derivative with respect to x, throughout each of

the following regions :

Ti: {a^y^x^b, Js}, T2. \a =i x =i y ^6, J,};

finally let 77 possess a discontinuity when x = y of the type

v_ "(¡^"Mr")-1-* Birkhoff, these Transactions, vol. 9 (1908), p. 377 ; Bounitzky, L i o u v i 11 e ' s

Journal, ser. 6, vol. 5 (1909), p. 65 ; Bôcher, Annals of Mathematics,

ser. 2, vol. 13 (1911-12), p. 71.



We use the notation 77 ($*'") to mean lim77, (?*'•') e > 0; also

tf(V) = lim77(^ ')•\2/± / e¿0 \íc±e /

Because of the continuity throughout 7i and 72, it is clear that

Replacing u by its value from (1), we find

lm = [h(1_')-b(1 + ')]m*,.)

+Í dx+ 4>(x,s)H^yS^ X(w, s)dy

+mâûR')]^.^-Hence, on account of (2), u as given by (1) satisfies (A) for every continuous

function X ( x, s ) if and only if 77 (*') and G (xy't) respectively satisfy the

equations

(3)

(4)

(p(x,s)H\^ySJ=Q,

'[•CO]—(.0*0-Both of these equations have to be considered separately in the regions 7"i

and 72, although the formal work is the same.

We may now regard the functions

*(:+,,)''(r"M:0as the initial functions given at a fixed point y in the interval Ix. These

functions will at present be assumed to be continuous in their respective

variables, and to satisfy condition (2); otherwise they are arbitrary, pending

further determination.

By (2), § 2, the solution of (3) is

<« *(;•)-*(;*)*(;■*■•)•

in which the ± signs correspond respectively to the regions 7i, 7*2.



The equation (4) may be solved by the result of § 2 in the form

<7,)--<7Hy;thr <::)<::>-m7HM7hr<7MM7hh-

By Corollary III, Theorem I, § 2, this simplifies into

<7)-<7Hïhl'i7My>+jT-(;ïW)*

Replacing 77 ( £ ' ) by its values from (5) and making use of the definition

(10), § 2, this further simplifies into

<7)-<7)<l')(6)

+r-c:K;)*+*(r *)•(;:)■where the ± signs again correspond to the regions 7i and T2. It is important

to observe that the function G ( xv \ ) thus determined is continuous through-

out IxyJst, because the only possible place of discontinuity is when x = y,

but then S ( xy \ ) = 0 by virtue of its definition.

We are now to determine H (%*'") and G (vy\) so that the expression (1)

also satisfies the boundary equation ( Bo ) for all X ( x, s ). Upon substitu-

tion of (1) in U[u] we have

U[u]= £[a(s)H^yS^ + ß(s)H(byS^y(y,s)dy

+fi><>K:v<">"(r)+ u[G(ySt^\(y,t)dtdy.

We shall have U[u] = 0 for all X(x, s) if and only if the equations

(7) a(S)Jí(°S) + í(»)H(^).0.

(8> v[a(7)]+M")H(7)+Bi,''>H(l')-0are satisfied.



On substituting in (7) for H(l') and H(by') their values from (5), w^

obtain an equation in 77 (yy+'") and 77 (\""■*), which together with (2) enables

us to find for these functions the values

(9) H[y+'s) = —t-\> H( )=-t—\ — >\y ) g(y,s) \y ) g(y,s)

since we confine ourselves to the case g(y, s) 4= 0 (§ 4). It is convenient

at this stage to introduce the following abbreviations which will be useful

later.

gi(y,s) = a(s)RâySJ, g2(y,s) = ß(s)R[by '),

<:)=^'H^+ßMlst)+lß^^<:yThus according to the notations (6), § 3, we have*

(11) g(y,s)=gi(y,s)+g2(y,s), G^y *) = 6^y j) 4- G2 (y *),

and the equations (9) may be written

(90 H(y+>S)=9-^, *(*"'•) ...ftílilí\y ) g(y,s) \y J g(y,s)

Now from the equation (6) we have

+r«(;:M;r)*+*(r'*W;;)-•C:)—(î')-(ï:)

_ +rc)o+*(r'')o* Note that G ( j and G ( 1 are two entirely different functions.


1918]

whence

LINEAR INTEGRO-DIFFERENTIAL EQUATIONS 401

"K;:)]-»<»"«(;;)*r*(,:W;0*+*(r,')[«w(;;)+r^'">a(;i)*]

«(;+,,)H'C0+r«-'>'(;rH-We shall now substitute this value in (8) and also replace 77 ( v ' ), 77 ( J ' )

by their values from (5). If in the resulting equation we replace 77 (yy+''),

77 (y~'e) by their values from (9'), we find that (8) reduces to

-ijh)[<"i!,',)a'(7t)-sii!''t)a'(7>)l

•(;ï)-'(.ï)+r*o(:o*-or

(12)

if we write for short

(13) F(,;)-

-1

g(y,s)g(y,t)

giiy.t) Gi(yst)

92(y,t) Gi(yst)

The kernel K ( „ ' ) is the same as that in the equation (7'), § 3.

Now if the homogeneous system (^40, -Bo) is incompatible, then the kernel

K(y'r) possesses a resolvent function Q ( y ' ) and the equation (12) pos-

sesses a unique solution given by

<"> <7)-<7tyi:<7H7>-Definition. The functions 77(*ä), G (»O are said to form a system

of Green's functions of the integro-differential boundary problem (A0, B0):

L[u] = 0, U[u] = 0, where U[u] is assumed to be integro-linearly self-

independent and subject to the condition ( C ), if they are defined respectively

in the regions Ixy J8 and Ixy J,t and possess the following properties:

1. H(xy') is continuous together with the first partial derivative with

respect to x in the regions Ti and T2, and

»(ro-'G-")-1-License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


2. G ( y « ) is continuous together with the first partial derivative with

respect to x throughout the region Ixy Jst.

3. Throughout 7i and T2 the functions II (I') satisfies the equations

(3) and (7).

4. The function G(îî) satisfies the equations (4) and (8).

Theorem I. When Green's functions exist, the semi-homogeneous system

(A, Bo) possesses a solution given by the formula (1).

We have seen in the above deduction that Green's functions exist if the

system (Ao,Bo) is incompatible. Because of the fact (Theorem III, §4)

that when (A0, B0) is compatible not every semi-homogeneous system (A,

Bo ) can have a solution, it follows that Green's functions do not exist for this

case. Hence

Theorem II. A necessary and sufficient condition that a system of Green's

functions exist for a system (Ao, B0), in which U is self-independent and (C)

is fulfilled, is that the system (Ao, Bo) be incompatible. When this condition

is satisfied, the solution given by (1) is the unique solution.

The last fact follows from the Corollary to Theorem III, § 3. From the

theorem just stated, it follows that the equation (12) cannot possess a solution

whenever the system (Ao, Bo) is compatible. Thus we have the

Corollary. When a system (Ao, B0), in which U is self-independent and

fulfills (C), is compatible, the function (13) cannot vanish identically; and

does not vanish identically for every (pi(y, s) which satisfies

4>(y,s) = j 4>(y,r)K^yrs^dr.

Theorem III. For a system (A, B0) there cannot exist more than one set

of functions H(xys), G(y\) such that (1) is a solution of the system for every

\(x, s); and if such a set exists, it consists of the Green's functions for the

system.

When the reduced system (A0, B0) is compatible, no such functions 77

and G can exist, because in that case not every semi-homogeneous system

(A, Bo) can have a solution (Theorem III, § 4). When (î0, B0) is incom-

patible, Green's functions exist and (1) is the unique solution of (A, Bo).

Hence if there exists another set of functions, H' and G', such that

u(x,s) = £h' (**)m2/ , >)dy + £ £'g'(**)x(? ,t)dtdy

is also a solution of (A, Bo), this solution must be identical with (1) and



therefore the difference of this and (1) is identically zero. Since X(a;, s) is

arbitrary, we find, by using the lemma in § 5, 77' = 77, G' = G.

. If in the system (A, Bo) we replace the boundary condition U = 0 by

another boundary condition U' = 0, where U' is an integro-linear function

of U, then (1) will be obviously also a solution of the resulting system; hence

Corollary. The Green's functions of a system are invariant of the choice

of boundary conditions, provided the different choices of boundary expressions

are integro-linearly connected.

Another important property is that there exists a symmetrical relation

between the Green's functions of the given system and the adjoint system.

From Corollary I, Theorem I, § 6, it follows that the adjoint boundary con-

dition V = 0 is self-independent. By reference to Theorems II, III, § 7,

we infer from Theorem II :

Theorem IV. If the system (A0, B0) possesses Green's_functions, 77, G,

the adjoint system (Ao, B0) possesses Green's functions, 77, G.

The solution of the adjoint semi-homogeneous system

(I,Bo) - M[v}= u(x,s), V[v] = 0

is given by

(15) v(x,s) = £ h(XS^ u(y,s)dy + £ £ G(XySty(y,t)dtdy.

Let u(x,s) be the solution of the system (A, B0) given by (1). Then, by

Green's theorem,

[«(a:,*)X(a;,i) + u(x, s)u(x, s)]dsdx = 0.■

On the substitution of the values of u and v from (1) and (15), we have

££f.t[B(7)+H(7)hx'')"ii>-°)d*d>dx

+£ ¡y? i"[K7)+G(ll)]H*-s)>'{>-t)d'did»dx-0-

which holds for every X and u. Hence, by the lemma in § 5,

(1e, *(;•)--*(:')■ o-«(:.')•Theorem V. The Green's functions of adjoint systems satisfy (16).

Theorem VI. If two systems

L[u]=\(x,s), , L'[u]=\(x,s),{A'Bo) Ui[u]=0, {A'Ba) U[[u) = Q



have the same Green's functions H(*'), G(xy"t), and if the adjoint system

(A0, B0) exists, then the expressions L and L' are identical and U[ is an integro-

linear function of Ui.

Since the Green's functions are the same for both systems, the function

formed from them by the formula (1) satisfies both systems, hence it satisfies

the homogeneous equation

L[u] - L'[u] = 0,that is,

(p"(x,s)u(x,s)+ i V'{XSAu(x,t)dt = 0,

if we let4>"(x,s) = 4>(x,s) - 4>'(x, s),

If we substitute (1) in this equation, we find

Ça(p"(x,s)H^XyS^\(y,s)dy

+nv<" ><K::)+*"U)»(r)

By the lemma of § 5, we obtain

*"(*,*)#(**) «0,

(17)

Let us take the limit of the first of these equations as y approaches x first

from above and then from below. This gives

(p"(x,s)h(kXx±S^ =0,

and by subtracting one of these equations from the other, we see from (2)

that d>" = 0.

Substituting thisjvalue in the second equation (17), and replacing 77 and G

by their values — 77, — G, we find



Hence

Now the first member of this equation is, by (15), the solution of the system

-M[v] = r[ySt), Vi[v] = Q,

regarded as equations in y and t. Hence \¡/" = 0. This completes the proof

that L[u] and L'[u] are identical.

Our theorem will be proved if we can show that Ui and U[ are integro-

linearly connected. For this purpose we substitute in U{ for u(a, s) and

u(b, s) their values from (14), § 6. This gives

U'As) = Mi(s)Ui(s)+f Ni(s,t)Ui(t)i

+ M2(s)Ui(s)+j Ni(s,t)Ui(t)dt,

\dt-0

in which

Mi(s) = - a\(s)yi(s) -r-ß'As)h(s),

Ni(s,t) = - a'As)Ci(t,s)+ß[(s)DAt,s) -A'As,t)yAt)

+ B'As,t)5i(t)+f [-A'As,r)Ci(t,r)

+ B[(s,r)Di(t,r)]dr.

And if we can show that M2(s) = 0, N2 (s, t ) = 0, we shall have established

an integro-linear relation between Ui and U{.

To prove M2 ( s ) = 0, we make use of (7) and the corresponding formula

for (A', B'a). We have, since (A, Bo) and (A', B'0) have the same Green's

functions,

*<.>*(") + A(0*(J*)-a,

«i(.)ff(j') + Ä(Off(*')-o.

Now for each constant value *0 the functions 77(^ '") and 77(J *°) cannot

both vanish identically, because otherwise we would have from (5) both

H (vy~''°) and 77(*+' '") identically zero, which is impossible owing to the dis-

continuity of 77. Consequently

-a[(s)ßi(s) +ß\(s)ai(s) ^0.

Hence, from (8), § 6, we have M2 = 0.



To show Ni ( s, t ) = 0, we have from the formulée corresponding to (8) in

the case of the systems (A, B0) and (A', B¡,)

-r-[9(::)]-a«..)*C')+i».(«..)t(i').

The subscript 1 has been dropped from Î7i and Fi for convenience, and the

variable subscripts are inserted to indicate the variables operated on. On

account of the relation (16) we have the identity

which may be written

U'x.[c1(t,s)Ë(ax*} + D1(t,s)H(bxS}]

+ Vvt[A'l(s,t)H(ayt^ + B[(s,t)H(byt^ = Q.

Expanding and collecting terms, we find

[«;(,)//(: + 5)-.3;(,)7/(^)]ci(i,S)4-[a;(S)77(^)

+ ß\(s)ü(hb_S)]l)i(t,s) + [yi(t)H(aa + t^

+ 5i(0/7(^)].i;(,,0+[7i(077(^¿)

+ oi(t)H^bb_t^B{(s,t)+£Â[(s,r)Ci(t,r)^Hâa + r^

+B(l+')]+B;i.,r)D,«,r)[a(l_')+a(l_<)]}är

+jf{4(..r,a„,„[*C')+,(;')]

+ B'1(s,r)Ci(t,r)[ll(abr} + H(bar^dr = 0.

By means of the relations



and (7), (16), (4), the first member of this equation reduces precisely to the

expression N2(s, t). Thus our proof is completed.

Corollary. A necessary and sufficient condition that the Green's functions

of a system be skew-symmetric, i. e.,

*(;•)-*(:•)• <7,)--°(i:yis that the integro-differential expression L [ u ] be anti-self-adjoint and the

boundary condition U [ u ] = 0 be self-adjoint.

The sufficiency of this theorem follows from Theorem IV, and the necessity

from the theorem just proved.

Harvard University

May, 1917


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